SMHS BiasPrecision - Revision history

Dinov: /* Application II: Uniform θ-Estimate Experiment */

2014-09-03T13:54:20Z

‎Application II: Uniform θ-Estimate Experiment

Dinov: /* Application II: Uniform θ-Estimate Experiment */

2014-09-03T13:53:26Z

‎Application II: Uniform θ-Estimate Experiment

Dinov: /* Application II: Uniform θ-Estimate Experiment */

2014-09-03T13:53:07Z

‎Application II: Uniform θ-Estimate Experiment

Dinov: /* Application II: Uniform θ-Estimate Experiment */

2014-09-03T13:51:24Z

‎Application II: Uniform θ-Estimate Experiment

Dinov: /* Bias */

2014-09-03T13:51:03Z

‎Bias

Dinov: /* Bias */

2014-09-03T13:50:10Z

‎Bias

Dinov: /* Bias */

2014-09-03T13:49:50Z

‎Bias

Dinov: /* Bias */

2014-09-03T13:49:16Z

‎Bias

Liyufang: /* References */

2014-08-29T16:58:42Z

‎References

Liyufang: /* References */

2014-08-29T16:58:23Z

‎References

@@ Line 54: / Line 54: @@
 : The $ θ $-estimate experiment allows us to generate a random sample $ X_1,X_2,…,X_n $ of size $ n$ from the uniform distribution on (0,1). The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded:
-: $ U=$ the minimum n for which the sum $=X_1+X_2+⋯+X_n >1.$  That is,$ U=\arg\min_n {X_1+X_2+⋯+X_n >1} $, note that all $X_i\geq 0 $ so such $n$ exists.
+: $ U=$ the minimum n for which the sum $=X_1+X_2+⋯+X_n >1.$  That is,$ U=\arg\min_n {\{ X_1+X_2+⋯+X_n >1\} } $, note that all $X_i\geq 0 $ so such $n$ exists.
 *Goal: The purpose of the Uniform E-Estimate Experiment is to provide an interactive computer demonstration illustrating a simple idea behind a stochastic simulation for estimating the natural number e. If U = minimum n for which the sum $ S=X_1+X_2+⋯+X_n>1 $, then the expected value of $ U,E(U) $, is approximately equal to the natural number $ e~2.7182.... $

@@ Line 54: / Line 54: @@
 : The $ θ $-estimate experiment allows us to generate a random sample $ X_1,X_2,…,X_n $ of size $ n$ from the uniform distribution on (0,1). The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded:
-$ U=$ the minimum n for which the sum $=X_1+X_2+⋯+X_n >1.$  That is,$ U=\argmin_n {X_1+X_2+⋯+X_n >1} $, note that all $X_i\geq 0 $ so such $n$ exists.
+: $ U=$ the minimum n for which the sum $=X_1+X_2+⋯+X_n >1.$  That is,$ U=\arg\min_n {X_1+X_2+⋯+X_n >1} $, note that all $X_i\geq 0 $ so such $n$ exists.
 *Goal: The purpose of the Uniform E-Estimate Experiment is to provide an interactive computer demonstration illustrating a simple idea behind a stochastic simulation for estimating the natural number e. If U = minimum n for which the sum $ S=X_1+X_2+⋯+X_n>1 $, then the expected value of $ U,E(U) $, is approximately equal to the natural number $ e~2.7182.... $

@@ Line 54: / Line 54: @@
 : The $ θ $-estimate experiment allows us to generate a random sample $ X_1,X_2,…,X_n $ of size $ n$ from the uniform distribution on (0,1). The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded:
-$ U=minimum\, n\, for\, which\, the\, sum S=X_1+X_2+⋯+X_n>1. $  That is,$ U=argmin_n (X_1+X_2+⋯+X_n>1) $, note that all $X_i≥0 $ so such n exists.
+$ U=$ the minimum n for which the sum $=X_1+X_2+⋯+X_n >1.$  That is,$ U=\argmin_n {X_1+X_2+⋯+X_n >1} $, note that all $X_i\geq 0 $ so such $n$ exists.
 *Goal: The purpose of the Uniform E-Estimate Experiment is to provide an interactive computer demonstration illustrating a simple idea behind a stochastic simulation for estimating the natural number e. If U = minimum n for which the sum $ S=X_1+X_2+⋯+X_n>1 $, then the expected value of $ U,E(U) $, is approximately equal to the natural number $ e~2.7182.... $

@@ Line 53: / Line 53: @@
 *Description: This experiment is used to estimate the value of the natural number using simulation.
-The $ θ $-estimate experiment allows us to generate a random sample $ X_1,X_2,…,X_n $ of size $ n$ from the uniform distribution on (0,1). The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded:
+: The $ θ $-estimate experiment allows us to generate a random sample $ X_1,X_2,…,X_n $ of size $ n$ from the uniform distribution on (0,1). The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded:
 $ U=minimum\, n\, for\, which\, the\, sum S=X_1+X_2+⋯+X_n>1. $  That is,$ U=argmin_n (X_1+X_2+⋯+X_n>1) $, note that all $X_i≥0 $ so such n exists.
@@ Line 60: / Line 60: @@
 *Application: Estimation of the natural number $ e $ is very important in many science and technology developments and studies. There are deterministic algorithms as well as stochastic methods for estimating the value of $ e $ . Many of these provide up to 10-billion decimal place accuracy for $ e $ .
-This experiment demonstrates an easy to understand, demonstrate and utilize protocol for a stochastic estimation of $ e $ . The algorithm may be significantly improved in terms of both speed of convergence and accuracy, relative to sample size $ (n) $.However, the emphasis in this experiment is simplicity and simulation of a transcendental number in real-time using basic tools (sampling from uniform distribution).
+: This experiment demonstrates an easy to understand, demonstrate and utilize protocol for a stochastic estimation of $ e $ . The algorithm may be significantly improved in terms of both speed of convergence and accuracy, relative to sample size $ (n) $.However, the emphasis in this experiment is simplicity and simulation of a transcendental number in real-time using basic tools (sampling from uniform distribution).
 *[http://link.springer.com/article/10.1023/A:1009982611386 This article] compared bias and precision statistics in regression analysis when measurement techniques are compared, it also compared the inconsistencies occurred in reporting the results of this form of analysis in cardiac output measurement. It performed a MEDLINE search dating from 1986 and surveyed studies comparing techniques of cardiac output measurement using bias and precision statistics. This paper constructed an error-gram from the percentage error in the test and reference methods and used the error-gram to determine acceptable limits of agreement between methods. It came to the conclusion that when using bias and precision statistics, cardiac output, bias, limits of agreement, and percentage error should be presented and argued that acceptance of a new technique should rely on limits of agreement of up to ± 30% using current reference methods.

@@ Line 23: / Line 23: @@
 * In statistical computing, when we estimate a model parameter $θ$ we typically collect a dataset $X$ and estimate the probability distribution of the observed data given the parameter: $P_\theta(x) = P(x\mid\theta)$. We usually estimate $\hat\theta$ using the observed data and hope that $\hat\theta \approx \theta$. Then the '''bias''' of the estimator ($\hat\theta$) is:
 : $$\text{Bias}_\theta[\,\hat\theta\,] = E_\theta[\,\hat{\theta}\,]-\theta = E_\theta[\, \hat\theta - \theta \,],$$
-where $E_\theta$ is the expected value of the (unknown) population parameter $\theta$ over the distribution $P_\theta(x) = P_\theta(x\mid\theta)$. If we were able to observe all possible outcomes $x$ and average the corresponding $\hat\theta$ estimates we would get $E_\theta$.
+where $E_\theta$ is the expected value of the (unknown) population parameter $\theta$ over the distribution $P_\theta(x) = P_\theta(x\mid\theta)$. If we were able to observe all possible outcomes $x$ and average the corresponding $\hat\theta$ estimates we would get $\theta$, otherwise the estimate is biased.
 ====Precision====

@@ Line 21: / Line 21: @@
 :(3) confounding bias: differences between cases and controls or exposed and unexposed distorts your estimates of the truth.
 *Also, the internal validity of a test may also be violated because of pure chance, the luck of the draw gets you a study sample that is not representative of the larger population. If a test is not internal valid, it is not external valid and cannot be generalized to any one.
-* In statistical computing, when we estimate a model parameter $θ$ we typically collect a datawset $X$ and estimate the probability distribution of the observed data given the parameter: $P_\theta(x) = P(x\mid\theta)$. We usually estimate $\hat\theta$ using the observed data and hope that $\hat\theta \approx \theta$. Then the '''bias''' of the estimator ($\hat\theta$) is:
+* In statistical computing, when we estimate a model parameter $θ$ we typically collect a dataset $X$ and estimate the probability distribution of the observed data given the parameter: $P_\theta(x) = P(x\mid\theta)$. We usually estimate $\hat\theta$ using the observed data and hope that $\hat\theta \approx \theta$. Then the '''bias''' of the estimator ($\hat\theta$) is:
-: $$\text{Bias}_\theta[\,\hat\theta\,] = \{E}_\theta[\,\hat{\theta}\,]-\theta = \{E}_\theta[\, \hat\theta - \theta \,],##
+: $$\text{Bias}_\theta[\,\hat\theta\,] = E_\theta[\,\hat{\theta}\,]-\theta = E_\theta[\, \hat\theta - \theta \,],##
-where $\{E}_\theta$ is the expected value of the (unknown) population parameter $\theta$ over the distribution $P_\theta(x) = P_\theta(x\mid\theta)$. If we were able to observe all possible outcomes $x$ and average the corresponding $\hat\theta$ estimates we would get $\{E}_\theta$.
+where $E_\theta$ is the expected value of the (unknown) population parameter $\theta$ over the distribution $P_\theta(x) = P_\theta(x\mid\theta)$. If we were able to observe all possible outcomes $x$ and average the corresponding $\hat\theta$ estimates we would get $E_\theta$.
 ====Precision====

← Older revision		Revision as of 16:58, 29 August 2014
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	*[http://www.astm.org/ILS/precisionbias.html Precision and Bias]		*[http://www.astm.org/ILS/precisionbias.html Precision and Bias]
−	*[http://www.astm.org/SNEWS/MARCH_2000/P&B_mar00.html Facts vs. Fiction: The Truth About Precision and Bias~~, by~~ Pat Picariello]	+	*[http://www.astm.org/SNEWS/MARCH_2000/P&B_mar00.html Facts vs. Fiction: The Truth About Precision and Bias / Pat Picariello]

@@ Line 138: / Line 138: @@
 ===References===
-*[http://mirlyn.lib.umich.edu/Record/004199238 Statistical inference / Casella,G. & Berger,R.]
+*[http://www.astm.org/ILS/precisionbias.html  Precision and Bias]
-*[http://mirlyn.lib.umich.edu/Record/004232056 Sampling / Thompson, S. K.]
+*[http://www.astm.org/SNEWS/MARCH_2000/P&B_mar00.html  Facts vs. Fiction: The Truth About Precision and Bias, by Pat Picariello]
-*[http://mirlyn.lib.umich.edu/Record/004133572 Sampling Theory and Methods /  Sampath, S.]
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